Optimal Universal Quantum Circuits for Unitary Complex Conjugation

Let U-d be a unitary operator representing an arbitrary d-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number k of calls of U-d into its complex conjugate (U-d) over bar. Our circuits admit a parallel implementation and are proven to be optimal for any k and d with an average fidelity of ?F ? = k+1/d(d-k). Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call (k = 1) of the operation Ud, and the special case of k = d - 1 calls. We then show that our results encompass optimal transformations from k calls of U-d to f (U-d) for any arbitrary homomorphism f from the group of d-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.